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versary. To do honour to the occasion, Lobatchewsky attended
the exercises in person to present a copy of his Pangeometry, the
completed work of his scientific life. This work (in French and
Russian) was not written by his own hand, but was dictated, as
Lobatchewsky was now blind. A few months later he died, on
24 February 1856, at the age of sixty-two.

To see what Lobatchewsky did we must first glance at
Euclid's outstanding achievement. The name Euclid until quite
recently was practically synonymous with elementary school
geometry. Of the man himself very little is known beyond his
doubtful dates, 330-275 B.C. In addition to a systematic
account of elementary geometry his Elements contain all that
was known in his time of the theory of numbers. Geometrical
teaching was dominated by Euclid for over 2,200 years. His
part in the Elements appears to have been principally that of a
co-ordinator and logical arranger of the scattered results of his
predecessors and contemporaries, and his aim was to give a
connected, reasoned account of elementary geometry such that
every statement in the whole long book could be referred back
to the postulates. Euclid did not attain this ideal or anything
even distantly approaching it, although it was assumed for
centuries that he had.

Euclid's title to immortality is based on something quite
other than the supposed logical perfection which is still some-
times erroneously ascribed to him. This is his recognition that
the fifth of his postulates (his Axiom XI) is a pure assumption.


The fifth postulate can be stated in many equivalent forms,
each of which is deducible from any one of the others by means
of the remaining postulates of Euclid's geometry. Possibly the
simplest of these equivalent statements is the following: Given
any straight line I and a point P not on 19 then in the plane
determined by I and P it is possible to draw precisely one straight